The Cauchy problem for a parabolic p-Laplacian equation with combined nonlinearities

Abstract

This article studies the Cauchy problem for the evolution p-Laplacian equation ut−Δpu=λum+μ|∇u|qur in RN×(0,T). The local existence, global existence and nonexistence of solutions are investigated. In particular, for the case λ>0 and μ>0, we obtain an optimal Fujita-type result, which demonstrates the positive gradient term brings about the discontinuity phenomenon of the critical exponent. For the case λ>0 and μ<0, the existence and nonexistence of global solutions are also discussed.